Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 0.7

Time: 2.5m

Precision: 64

• could not determine a ground truth for program body

  1. z = -8.778157517572568e+212

Math

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

↓

\[\left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)}\right) + \frac{\mathsf{fma}\left(-1259.139216722402807135949842631816864014, \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 - z\right)\right)\right), \left(2 - z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({0.9999999999998099298181841732002794742584}^{3} + {\left(\frac{676.5203681218850988443591631948947906494}{1 - z}\right)}^{3}, \left(-z\right) + 4, \mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot -176.6150291621405870046146446838974952698\right), 3 - z, \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot 771.3234287776531346025876700878143310547\right), \left(\left(-z\right) + 7\right) \cdot \left(6 - z\right), \mathsf{fma}\left(9.984369578019571583242346146658263705831 \cdot 10^{-6}, 6 - z, \left(\left(-z\right) + 7\right) \cdot -0.1385710952657201178173096423051902092993\right) \cdot \left(\left(3 - z\right) \cdot \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right)\right)\right)\right)}{\left(2 - z\right) \cdot \left(\left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 - z\right)\right)\right)\right)}\right) \cdot \frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{0.5 + \left(\left(-z\right) + 7\right)}}\]

C

double f(double z) {
        double r157301 = atan2(1.0, 0.0);
        double r157302 = z;
        double r157303 = r157301 * r157302;
        double r157304 = sin(r157303);
        double r157305 = r157301 / r157304;
        double r157306 = 2.0;
        double r157307 = r157301 * r157306;
        double r157308 = sqrt(r157307);
        double r157309 = 1.0;
        double r157310 = r157309 - r157302;
        double r157311 = r157310 - r157309;
        double r157312 = 7.0;
        double r157313 = r157311 + r157312;
        double r157314 = 0.5;
        double r157315 = r157313 + r157314;
        double r157316 = r157311 + r157314;
        double r157317 = pow(r157315, r157316);
        double r157318 = r157308 * r157317;
        double r157319 = -r157315;
        double r157320 = exp(r157319);
        double r157321 = r157318 * r157320;
        double r157322 = 0.9999999999998099;
        double r157323 = 676.5203681218851;
        double r157324 = r157311 + r157309;
        double r157325 = r157323 / r157324;
        double r157326 = r157322 + r157325;
        double r157327 = -1259.1392167224028;
        double r157328 = r157311 + r157306;
        double r157329 = r157327 / r157328;
        double r157330 = r157326 + r157329;
        double r157331 = 771.3234287776531;
        double r157332 = 3.0;
        double r157333 = r157311 + r157332;
        double r157334 = r157331 / r157333;
        double r157335 = r157330 + r157334;
        double r157336 = -176.6150291621406;
        double r157337 = 4.0;
        double r157338 = r157311 + r157337;
        double r157339 = r157336 / r157338;
        double r157340 = r157335 + r157339;
        double r157341 = 12.507343278686905;
        double r157342 = 5.0;
        double r157343 = r157311 + r157342;
        double r157344 = r157341 / r157343;
        double r157345 = r157340 + r157344;
        double r157346 = -0.13857109526572012;
        double r157347 = 6.0;
        double r157348 = r157311 + r157347;
        double r157349 = r157346 / r157348;
        double r157350 = r157345 + r157349;
        double r157351 = 9.984369578019572e-06;
        double r157352 = r157351 / r157313;
        double r157353 = r157350 + r157352;
        double r157354 = 1.5056327351493116e-07;
        double r157355 = 8.0;
        double r157356 = r157311 + r157355;
        double r157357 = r157354 / r157356;
        double r157358 = r157353 + r157357;
        double r157359 = r157321 * r157358;
        double r157360 = r157305 * r157359;
        return r157360;
}

↓

double f(double z) {
        double r157361 = 1.5056327351493116e-07;
        double r157362 = z;
        double r157363 = -r157362;
        double r157364 = 8.0;
        double r157365 = r157363 + r157364;
        double r157366 = r157361 / r157365;
        double r157367 = 12.507343278686905;
        double r157368 = 5.0;
        double r157369 = r157368 + r157363;
        double r157370 = r157367 / r157369;
        double r157371 = r157366 + r157370;
        double r157372 = -1259.1392167224028;
        double r157373 = 0.9999999999998099;
        double r157374 = 676.5203681218851;
        double r157375 = 1.0;
        double r157376 = r157375 - r157362;
        double r157377 = r157374 / r157376;
        double r157378 = r157377 - r157373;
        double r157379 = r157377 * r157378;
        double r157380 = fma(r157373, r157373, r157379);
        double r157381 = 4.0;
        double r157382 = r157363 + r157381;
        double r157383 = r157380 * r157382;
        double r157384 = 3.0;
        double r157385 = r157384 - r157362;
        double r157386 = 7.0;
        double r157387 = r157363 + r157386;
        double r157388 = 6.0;
        double r157389 = r157388 - r157362;
        double r157390 = r157387 * r157389;
        double r157391 = r157385 * r157390;
        double r157392 = r157383 * r157391;
        double r157393 = 2.0;
        double r157394 = r157393 - r157362;
        double r157395 = 3.0;
        double r157396 = pow(r157373, r157395);
        double r157397 = pow(r157377, r157395);
        double r157398 = r157396 + r157397;
        double r157399 = -176.6150291621406;
        double r157400 = r157380 * r157399;
        double r157401 = fma(r157398, r157382, r157400);
        double r157402 = 771.3234287776531;
        double r157403 = r157383 * r157402;
        double r157404 = fma(r157401, r157385, r157403);
        double r157405 = 9.984369578019572e-06;
        double r157406 = -0.13857109526572012;
        double r157407 = r157387 * r157406;
        double r157408 = fma(r157405, r157389, r157407);
        double r157409 = r157385 * r157383;
        double r157410 = r157408 * r157409;
        double r157411 = fma(r157404, r157390, r157410);
        double r157412 = r157394 * r157411;
        double r157413 = fma(r157372, r157392, r157412);
        double r157414 = r157394 * r157392;
        double r157415 = r157413 / r157414;
        double r157416 = r157371 + r157415;
        double r157417 = atan2(1.0, 0.0);
        double r157418 = r157417 * r157362;
        double r157419 = sin(r157418);
        double r157420 = r157417 / r157419;
        double r157421 = r157417 * r157393;
        double r157422 = sqrt(r157421);
        double r157423 = r157420 * r157422;
        double r157424 = 0.5;
        double r157425 = r157424 + r157387;
        double r157426 = r157363 + r157424;
        double r157427 = pow(r157425, r157426);
        double r157428 = r157423 * r157427;
        double r157429 = exp(r157425);
        double r157430 = r157428 / r157429;
        double r157431 = r157416 * r157430;
        return r157431;
}

Error

Bits error versus z

Derivation

1. Initial program 1.8

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

2. Simplified 2.0

   \[\leadsto \color{blue}{\left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)}\right) + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)} + \left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4}\right) + \frac{771.3234287776531346025876700878143310547}{3 + \left(-z\right)}\right) + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{-0.1385710952657201178173096423051902092993}{6 + \left(-z\right)}\right)\right)\right)\right) \cdot \frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{0.5 + \left(\left(-z\right) + 7\right)}}}\]
3. Using strategy rm

4. Applied frac-add 2.0

   \[\leadsto \left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)}\right) + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)} + \left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4}\right) + \frac{771.3234287776531346025876700878143310547}{3 + \left(-z\right)}\right) + \color{blue}{\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6} \cdot \left(6 + \left(-z\right)\right) + \left(\left(-z\right) + 7\right) \cdot -0.1385710952657201178173096423051902092993}{\left(\left(-z\right) + 7\right) \cdot \left(6 + \left(-z\right)\right)}}\right)\right)\right) \cdot \frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{0.5 + \left(\left(-z\right) + 7\right)}}\]

5. Applied flip3-+ 2.0

   \[\leadsto \left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)}\right) + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)} + \left(\left(\left(\color{blue}{\frac{{0.9999999999998099298181841732002794742584}^{3} + {\left(\frac{676.5203681218850988443591631948947906494}{1 - z}\right)}^{3}}{0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)}} + \frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4}\right) + \frac{771.3234287776531346025876700878143310547}{3 + \left(-z\right)}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6} \cdot \left(6 + \left(-z\right)\right) + \left(\left(-z\right) + 7\right) \cdot -0.1385710952657201178173096423051902092993}{\left(\left(-z\right) + 7\right) \cdot \left(6 + \left(-z\right)\right)}\right)\right)\right) \cdot \frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{0.5 + \left(\left(-z\right) + 7\right)}}\]

6. Applied frac-add 2.0

   \[\leadsto \left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)}\right) + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)} + \left(\left(\color{blue}{\frac{\left({0.9999999999998099298181841732002794742584}^{3} + {\left(\frac{676.5203681218850988443591631948947906494}{1 - z}\right)}^{3}\right) \cdot \left(\left(-z\right) + 4\right) + \left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot -176.6150291621405870046146446838974952698}{\left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot \left(\left(-z\right) + 4\right)}} + \frac{771.3234287776531346025876700878143310547}{3 + \left(-z\right)}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6} \cdot \left(6 + \left(-z\right)\right) + \left(\left(-z\right) + 7\right) \cdot -0.1385710952657201178173096423051902092993}{\left(\left(-z\right) + 7\right) \cdot \left(6 + \left(-z\right)\right)}\right)\right)\right) \cdot \frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{0.5 + \left(\left(-z\right) + 7\right)}}\]

7. Applied frac-add 0.7

   \[\leadsto \left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)}\right) + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)} + \left(\color{blue}{\frac{\left(\left({0.9999999999998099298181841732002794742584}^{3} + {\left(\frac{676.5203681218850988443591631948947906494}{1 - z}\right)}^{3}\right) \cdot \left(\left(-z\right) + 4\right) + \left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot -176.6150291621405870046146446838974952698\right) \cdot \left(3 + \left(-z\right)\right) + \left(\left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot 771.3234287776531346025876700878143310547}{\left(\left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(3 + \left(-z\right)\right)}} + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6} \cdot \left(6 + \left(-z\right)\right) + \left(\left(-z\right) + 7\right) \cdot -0.1385710952657201178173096423051902092993}{\left(\left(-z\right) + 7\right) \cdot \left(6 + \left(-z\right)\right)}\right)\right)\right) \cdot \frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{0.5 + \left(\left(-z\right) + 7\right)}}\]

8. Applied frac-add 2.0

   \[\leadsto \left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)}\right) + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)} + \color{blue}{\frac{\left(\left(\left({0.9999999999998099298181841732002794742584}^{3} + {\left(\frac{676.5203681218850988443591631948947906494}{1 - z}\right)}^{3}\right) \cdot \left(\left(-z\right) + 4\right) + \left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot -176.6150291621405870046146446838974952698\right) \cdot \left(3 + \left(-z\right)\right) + \left(\left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot 771.3234287776531346025876700878143310547\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 + \left(-z\right)\right)\right) + \left(\left(\left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(3 + \left(-z\right)\right)\right) \cdot \left(9.984369578019571583242346146658263705831 \cdot 10^{-6} \cdot \left(6 + \left(-z\right)\right) + \left(\left(-z\right) + 7\right) \cdot -0.1385710952657201178173096423051902092993\right)}{\left(\left(\left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(3 + \left(-z\right)\right)\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 + \left(-z\right)\right)\right)}}\right)\right) \cdot \frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{0.5 + \left(\left(-z\right) + 7\right)}}\]

9. Applied frac-add 2.0

   \[\leadsto \left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)}\right) + \color{blue}{\frac{-1259.139216722402807135949842631816864014 \cdot \left(\left(\left(\left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(3 + \left(-z\right)\right)\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 + \left(-z\right)\right)\right)\right) + \left(2 + \left(-z\right)\right) \cdot \left(\left(\left(\left({0.9999999999998099298181841732002794742584}^{3} + {\left(\frac{676.5203681218850988443591631948947906494}{1 - z}\right)}^{3}\right) \cdot \left(\left(-z\right) + 4\right) + \left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot -176.6150291621405870046146446838974952698\right) \cdot \left(3 + \left(-z\right)\right) + \left(\left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot 771.3234287776531346025876700878143310547\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 + \left(-z\right)\right)\right) + \left(\left(\left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(3 + \left(-z\right)\right)\right) \cdot \left(9.984369578019571583242346146658263705831 \cdot 10^{-6} \cdot \left(6 + \left(-z\right)\right) + \left(\left(-z\right) + 7\right) \cdot -0.1385710952657201178173096423051902092993\right)\right)}{\left(2 + \left(-z\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(3 + \left(-z\right)\right)\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 + \left(-z\right)\right)\right)\right)}}\right) \cdot \frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{0.5 + \left(\left(-z\right) + 7\right)}}\]

10. Simplified 0.7

    \[\leadsto \left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)}\right) + \frac{\color{blue}{\mathsf{fma}\left(-1259.139216722402807135949842631816864014, \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 - z\right)\right)\right), \left(2 - z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({0.9999999999998099298181841732002794742584}^{3} + {\left(\frac{676.5203681218850988443591631948947906494}{1 - z}\right)}^{3}, \left(-z\right) + 4, \mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot -176.6150291621405870046146446838974952698\right), 3 - z, \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot 771.3234287776531346025876700878143310547\right), \left(\left(-z\right) + 7\right) \cdot \left(6 - z\right), \mathsf{fma}\left(9.984369578019571583242346146658263705831 \cdot 10^{-6}, 6 - z, \left(\left(-z\right) + 7\right) \cdot -0.1385710952657201178173096423051902092993\right) \cdot \left(\left(3 - z\right) \cdot \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right)\right)\right)\right)}}{\left(2 + \left(-z\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099298181841732002794742584 \cdot 0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584 \cdot \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(3 + \left(-z\right)\right)\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 + \left(-z\right)\right)\right)\right)}\right) \cdot \frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{0.5 + \left(\left(-z\right) + 7\right)}}\]

11. Simplified 0.7

    \[\leadsto \left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)}\right) + \frac{\mathsf{fma}\left(-1259.139216722402807135949842631816864014, \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 - z\right)\right)\right), \left(2 - z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({0.9999999999998099298181841732002794742584}^{3} + {\left(\frac{676.5203681218850988443591631948947906494}{1 - z}\right)}^{3}, \left(-z\right) + 4, \mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot -176.6150291621405870046146446838974952698\right), 3 - z, \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot 771.3234287776531346025876700878143310547\right), \left(\left(-z\right) + 7\right) \cdot \left(6 - z\right), \mathsf{fma}\left(9.984369578019571583242346146658263705831 \cdot 10^{-6}, 6 - z, \left(\left(-z\right) + 7\right) \cdot -0.1385710952657201178173096423051902092993\right) \cdot \left(\left(3 - z\right) \cdot \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right)\right)\right)\right)}{\color{blue}{\left(2 - z\right) \cdot \left(\left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 - z\right)\right)\right)\right)}}\right) \cdot \frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{0.5 + \left(\left(-z\right) + 7\right)}}\]

12. Final simplification 0.7

    \[\leadsto \left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)}\right) + \frac{\mathsf{fma}\left(-1259.139216722402807135949842631816864014, \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 - z\right)\right)\right), \left(2 - z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({0.9999999999998099298181841732002794742584}^{3} + {\left(\frac{676.5203681218850988443591631948947906494}{1 - z}\right)}^{3}, \left(-z\right) + 4, \mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot -176.6150291621405870046146446838974952698\right), 3 - z, \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot 771.3234287776531346025876700878143310547\right), \left(\left(-z\right) + 7\right) \cdot \left(6 - z\right), \mathsf{fma}\left(9.984369578019571583242346146658263705831 \cdot 10^{-6}, 6 - z, \left(\left(-z\right) + 7\right) \cdot -0.1385710952657201178173096423051902092993\right) \cdot \left(\left(3 - z\right) \cdot \left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right)\right)\right)\right)}{\left(2 - z\right) \cdot \left(\left(\mathsf{fma}\left(0.9999999999998099298181841732002794742584, 0.9999999999998099298181841732002794742584, \frac{676.5203681218850988443591631948947906494}{1 - z} \cdot \left(\frac{676.5203681218850988443591631948947906494}{1 - z} - 0.9999999999998099298181841732002794742584\right)\right) \cdot \left(\left(-z\right) + 4\right)\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(\left(-z\right) + 7\right) \cdot \left(6 - z\right)\right)\right)\right)}\right) \cdot \frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{0.5 + \left(\left(-z\right) + 7\right)}}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- (- 1 z) 1) 7) 0.5) (+ (- (- 1 z) 1) 0.5))) (exp (- (+ (+ (- (- 1 z) 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.99999999999980993 (/ 676.520368121885099 (+ (- (- 1 z) 1) 1))) (/ -1259.13921672240281 (+ (- (- 1 z) 1) 2))) (/ 771.32342877765313 (+ (- (- 1 z) 1) 3))) (/ -176.615029162140587 (+ (- (- 1 z) 1) 4))) (/ 12.5073432786869052 (+ (- (- 1 z) 1) 5))) (/ -0.138571095265720118 (+ (- (- 1 z) 1) 6))) (/ 9.98436957801957158e-6 (+ (- (- 1 z) 1) 7))) (/ 1.50563273514931162e-7 (+ (- (- 1 z) 1) 8))))))